Taylor and Maclaurin Series - PowerPoint PPT Presentation

1 / 10
About This Presentation
Title:

Taylor and Maclaurin Series

Description:

Taylor and Maclaurin Series Lesson 9.10 Convergent Power Series Form Consider representing f(x) by a power series For all x in open interval I Containing c Then ... – PowerPoint PPT presentation

Number of Views:1123
Avg rating:3.0/5.0
Slides: 11
Provided by: SteveAr8
Category:

less

Transcript and Presenter's Notes

Title: Taylor and Maclaurin Series


1
Taylor and Maclaurin Series
  • Lesson 9.10

2
Convergent Power Series Form
  • Consider representing f(x) by a power series
  • For all x in open interval I
  • Containing c
  • Then

3
Taylor Series
  • If a function f(x) has derivatives of all orders
    at x c, then the seriesis called the Taylor
    series for f(x) at c.
  • If c 0, the series is the Maclaurin series for
    f .

4
Taylor Series
  • This is an extension of the Taylor polynomials
    from section 9.7
  • We said for f(x) sin x, Taylor Polynomial of
    degree 7

5
Guidelines for Finding Taylor Series
  • Differentiate f(x) several times
  • Evaluate each derivative at c
  • Use the sequence to form the Taylor coefficients
  • Determine the interval of convergence
  • Within this interval of convergence, determine
    whether or not the series converges to f(x)

6
Series for Composite Function
  • What about when f(x) cos(x2)?
  • Note the series for cos x
  • Now substitute x2 in for the x's

7
Binomial Series
  • Consider the function
  • This produces the binomial series
  • We seek a Maclaurin series for this function
  • Generate the successive derivatives
  • Determine
  • Now create the series using the pattern

8
Binomial Series
  • We note that
  • Thus Ratio Test tells us radius of convergence
    R 1
  • Series converges to some function in interval-1
    lt x lt 1

9
Combining Power Series
  • Consider
  • We know
  • So we could multiply and collect like terms

10
Assignment
  • Lesson 9.10
  • Page 685
  • 1 29 odd
Write a Comment
User Comments (0)
About PowerShow.com